Taking David Lewin's work as a point of departure, this essay uses geometry to reexamine familiar music-theoretical assumptions about intervals and transformations. Section 1 introduces the problem of "transportability," noting that it is sometimes impossible to say whether two different directions--located at two different points in a geometrical space--are "the same" or not. Relevant examples include the surface of the earth and the geometrical spaces representing n-note chords. Section 2 argues that we should not require that every interval be defined at every point in a space, since some musical spaces have natural boundaries. It also notes that there are spaces, including the familiar pitch-class circle, in which there are multiple paths between any two points. This leads to the suggestion that we might sometimes want to replace traditional pitch-class intervals with paths in pitch-class space, a more fine-grained alternative that specifies how one pitch class moves to another. Section 3 argues that group theory alone cannot represent the intuition that intervals have quantifiable sizes, proposing an extension to Lewin's formalism that accomplishes this goal. Finally, Section 4 considers the analytical implications of the preceding points, paying particular attention to questions about voice leading.
%0 Journal Article
%1 DmitriTymoczko09012009
%A Tymoczko, Dmitri
%D 2009
%J Journal of Music Theory
%K Musiktheorie Musik Intervall MaMu
%N 2
%P 227-254
%R 10.1215/00222909-2010-003
%T Generalizing Musical Intervals
%U http://jmt.dukejournals.org/cgi/content/abstract/53/2/227
%V 53
%X Taking David Lewin's work as a point of departure, this essay uses geometry to reexamine familiar music-theoretical assumptions about intervals and transformations. Section 1 introduces the problem of "transportability," noting that it is sometimes impossible to say whether two different directions--located at two different points in a geometrical space--are "the same" or not. Relevant examples include the surface of the earth and the geometrical spaces representing n-note chords. Section 2 argues that we should not require that every interval be defined at every point in a space, since some musical spaces have natural boundaries. It also notes that there are spaces, including the familiar pitch-class circle, in which there are multiple paths between any two points. This leads to the suggestion that we might sometimes want to replace traditional pitch-class intervals with paths in pitch-class space, a more fine-grained alternative that specifies how one pitch class moves to another. Section 3 argues that group theory alone cannot represent the intuition that intervals have quantifiable sizes, proposing an extension to Lewin's formalism that accomplishes this goal. Finally, Section 4 considers the analytical implications of the preceding points, paying particular attention to questions about voice leading.
@article{DmitriTymoczko09012009,
abstract = {Taking David Lewin's work as a point of departure, this essay uses geometry to reexamine familiar music-theoretical assumptions about intervals and transformations. Section 1 introduces the problem of "transportability," noting that it is sometimes impossible to say whether two different directions--located at two different points in a geometrical space--are "the same" or not. Relevant examples include the surface of the earth and the geometrical spaces representing n-note chords. Section 2 argues that we should not require that every interval be defined at every point in a space, since some musical spaces have natural boundaries. It also notes that there are spaces, including the familiar pitch-class circle, in which there are multiple paths between any two points. This leads to the suggestion that we might sometimes want to replace traditional pitch-class intervals with paths in pitch-class space, a more fine-grained alternative that specifies how one pitch class moves to another. Section 3 argues that group theory alone cannot represent the intuition that intervals have quantifiable sizes, proposing an extension to Lewin's formalism that accomplishes this goal. Finally, Section 4 considers the analytical implications of the preceding points, paying particular attention to questions about voice leading.},
added-at = {2013-02-02T14:42:13.000+0100},
author = {Tymoczko, Dmitri},
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bdsk-url-1 = {http://jmt.dukejournals.org/cgi/content/abstract/53/2/227},
bdsk-url-2 = {http://dx.doi.org/10.1215/00222909-2010-003},
biburl = {https://www.bibsonomy.org/bibtex/2fa1bf1e293d9c266710659f73d2e3c19/ks-plugin-devel},
date-added = {2011-08-16 11:37:57 +0200},
date-modified = {2011-08-16 11:39:45 +0200},
doi = {10.1215/00222909-2010-003},
eprint = {http://jmt.dukejournals.org/cgi/reprint/53/2/227.pdf},
file = {:Tymoczko/Generalizing Musical Intervals.pdf:PDF},
groups = {public},
interhash = {8fd035d28ad191894f40c60be40a475f},
intrahash = {fa1bf1e293d9c266710659f73d2e3c19},
journal = {Journal of Music Theory},
keywords = {Musiktheorie Musik Intervall MaMu},
number = 2,
pages = {227-254},
timestamp = {2013-02-02T14:42:13.000+0100},
title = {Generalizing Musical Intervals},
url = {http://jmt.dukejournals.org/cgi/content/abstract/53/2/227},
username = {keinstein},
verfuegbar = {Online},
volume = 53,
year = 2009
}